nLab canonical form

Idea

Systems of formal logic, such as type theory, try to transform expressions into a canonical form which then serves as the end result of the given computation or deduction. A formal system is said to enjoy canonicity if every expression reduces to canonical form.

More precisely, in type theory, a term belonging to some type is said to be of canonical form if it is explicitly built up using the constructors of that type. A canonical form is in particular a normal form (one not admitting any reductions), but the converse need not hold.

For example, the terms of type $\mathbb{N}$ (a natural numbers type) of canonical form are the numerals

$$S(S(S(\cdots (0)\cdots ))).$$

A type theory is said to enjoy canonicity if every closed term (i.e. a term in the empty context) computes to a canonical form. This is held to be an important meta-theoretic property of type theory, especially considered as a programming language or as a computational foundation for mathematics. It is related to normalization, which says that every open term can also be reduced to a “normal form” of some kind.

Canonicity vs axioms

Adding axioms to Martin-Löf type theory, such as the principle of excluded middle or the usual version of the univalence axiom, can destroy canonicity. The axioms result in “stuck terms” which are not of canonical form, yet neither can they be “computed” any further.

For instance, if we assume the law of excluded middle, then we can build a term $case(LEM(Goldbach),0,1) \colon \mathbb{N}$ which is $0$ or $1$ according as the Goldbach conjecture is true or false. Clearly this term doesn’t “compute”, but neither is it of canonical form (a numeral).

Similarly, using the univalence axiom, we can obtain a term $p : (\mathbf{2}=\mathbf{2})$ corresponding to the automorphism of the type $\mathbf{2}$ which switches $0_\mathbf{2}$ and $1_\mathbf{2}$. Then the term $transport(p,0_\mathbf{2})$ also has type $\mathbf{2}$, but doesn’t “compute” because the computer gets “stuck” on the univalence term.

On the other hand, this form of univalence can be given a “computational” interpretation while preserving canonicity in cubical type theory (see e.g. Huber 2019). It is conjectured that univalence can also be given a “computational” interpretation while preserving canonicity in higher observational type theory (see Altenkirch et al. 2024).

See also

• homotopy canonicity

References

Discussion of canonicity in univalent type theory is discussed in

• Simon Huber (with Thierry Coquand), Towards a computational justification of the Axiom of Univalence , talk at TYPES 2011 (pdf)

• Dan Licata, Robert Harper, Computing with Univalence (2012) (pdf)

• Dan Licata, Canonicity for 2-Dimensional Type theory (2011) (web)

• Simon Huber, Canonicity for Cubical Type Theory, Journal of Automated Reasoning, 63, 2019. (doi:10.1007/s10817-018-9469-1, arXiv:1607.04156)

• Thorsten Altenkirch, Yorgo Chamoun, Ambrus Kaposi, Michael Shulman, Internal parametricity, without an interval, Proceedings of the ACM on Programming Languages (POPL) 8 (2024) 2340-2369 [arXiv:2307.06448, doi:10.1145/3632920]